Let $R = \{(x, y) \in N \times N : \log_e(x + y) \leq 2\}$. Then the minimum number of elements, required to be added in $R$ to make it a transitive relation, is . . . . . . .

On set $A = \{1, 2, 3\}$, relations $R$ and $S$ are given by $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)\}$ and $S = \{(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)\}$. Then,

Let $N$ be the set of natural numbers and the relation $R$ on $N \times N$ is defined by $(a, b) R (c, d)$ if $ad(b + c) = bc(a + d)$. Then $R$ is:

Determine whether the following relation is reflexive,symmetric,and transitive:
Relation $R$ in the set $A$ of human beings in a town at a particular time is given by:
$R = \{(x, y) : x \text{ and } y \text{ live in the same locality}\}$

Let $R$ and $S$ be two equivalence relations on a set $A$. Then

Let $X = R \times R$. Define a relation $R$ on $X$ as: $(a_1, b_1) R (a_2, b_2) \Leftrightarrow b_1 = b_2$. Statement-$I$: $R$ is an equivalence relation. Statement-$II$: For some $(a, b) \in X$,the set $S = \{(x, y) \in X : (x, y) R (a, b)\}$ represents a line parallel to $y = x$. In the light of the above statements,choose the correct answer from the options given below: